Definition 3.3.14

Definition 3.3.14 (One-to-one functions). A function $f$ is one-to-one (or injective) if different elements map to different elements: $$ x\neq x'\implies f(x)\neq f(x') $$

Equivalently, a function is one-to-one if $$ f(x)=f(x')\implies x=x' $$

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Exercise 3.3.7

$\Rightarrow$ { $f$ is injective by C1 and Definition 3.3.14 }

$\Rightarrow$ { $g$ is injective by C2 and Definition 3.3.14 }

$\Rightarrow$ { Definition 3.3.14 }
Exercise 3.3.5

$\Rightarrow$ { By Definition 3.3.14 and A1: $\exists x_1,x_2\in X. (x_1\neq x_2)\land (f(x_1)=f(x_2))$ }

$\equiv$ { Definition 3.3.14 }

$\equiv$ { Definition 3.3.14 }
Exercise 3.3.3

$\equiv$ { Definition 3.3.14 }
Exercise 3.3.2

$\Rightarrow$ { A1 and Definition 3.3.14 }

$\Rightarrow$ { A2 and Definition 3.3.14 }

$\vdash$ { Definition 3.3.14 }